JEE 2023 — Mathematics PYQ

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Question

JEE 2023 — Mathematics PYQ

JEE | Mathematics | 2023

If all the six digit numbers $x_1x_2x_3x_4x_5x_6$ with $0 < x_1 < x_2 < x_3 < x_4 < x_5 < x_6$ are arranged in the increasing order, then the sum of the digits in the $72^{\text{th}}$ number is

Choose the correct answer:

Explanation

Solution:

Digits available: $\{1, 2, 3, 4, 5, 6, 7, 8, 9\}$

Starting with $1$ ( $x_1=1$ ): Chunaav $\binom{8}{5} = 56$ numbers.

Starting with $2$ ( $x_1=2$ ): Chunaav $\binom{7}{5} = 21$ numbers.

$56 + 21 = 77$ , jo $72$ se zyada hai. Matlab number $2$ se start hoga.

$72^{\text{th}}$ number = $77^{\text{th}} - 5$ positions peeche.

$77^{\text{th}}$ number ( $x_1=2$ ka aakhri): $256789$

$76^{\text{th}}$ : $246789$ , $75^{\text{th}}$ : $245789$ , $74^{\text{th}}$ : $245689$ , $73^{\text{rd}}$ : $245679$ , $72^{\text{th}}$ : $245678$

Sum of digits ( $245678$ ): $2+4+5+6+7+8 = 32$

Result: $32$